


Visual Verification

by Bitenomnom



Series: Mathematical Proof [33]
Category: Sherlock (TV)
Genre: Clothed Sex, Cuddling, Frottage, M/M, PWP, Porn, in!control!John, the power of demonstration
Language: English
Status: Completed
Published: 2012-11-06
Updated: 2012-11-06
Packaged: 2017-11-18 02:42:27
Rating: Explicit
Warnings: Creator Chose Not To Use Archive Warnings
Chapters: 1
Words: 2,715
Publisher: archiveofourown.org
Story URL: https://archiveofourown.org/works/555988
Author URL: https://archiveofourown.org/users/Bitenomnom/pseuds/Bitenomnom
Summary: <blockquote class="userstuff">
              <p>John pulled Sherlock’s face down and leaned heavier against him to whisper in his ear. “I said scientific rigor,” John told him. “I <i>meant</i> a demonstration, not a discussion over whether bloody Scotland Yard was going to figure out the meaning of your convoluted description.”</p>
            </blockquote>





	Visual Verification

**Author's Note:**

> Some porn to start off the week, hooray! I keep thinking one day I'm going to mix it up and actually write some graphic anal sex or something, but, er, what can I say...I am sort of really a fan of frottage.

When performing linear or multilinear regression on data to form a model, we need to make sure that the model satisfies a few assumptions, or the regression will be meaningless. The best way to check for the qualities we need to see is often to check visually, since it is then easy to determine what measures need to be taken to fix the data.

 

First, we need to check a model for non-normality (see [Normality](http://archiveofourown.org/works/510920)). Namely, we need to make sure that the residuals have a normal distribution. This can be done by looking at a Q-Q plot: if it is a straight line, you have normality. If not, then you must attempt to perform some type of transformation on the y-variable in order to straighten it out.

 

Next, we must check for non-constant variance (see [Variance](http://archiveofourown.org/works/505398)); that is, for our model to work, we want to make sure that the variance is same for the entire range of the data (for instance, it would be bad if at the higher end of your x scale, the y value is all over the place, while at the lower end the values are all pretty clustered together). This can be checked for by plotting the studentized residuals (see [The Cost of Decreased Variance](http://archiveofourown.org/works/551771)) vs. the value of each variable in turn—you shouldn’t see any particular trend (i.e. a positively-sloped linear pattern, which would indicate that variance gets bigger as the value of one of your variables gets bigger). It is difficult to transform your way out of non-constant variance. One way of fixing this problem is to use weighted least squares.

 

Finally, we must check for non-linearity. Naturally, this is one requirement we have for doing a linear or multilinear regression. The way to check for this problem visually is to use a “component and residual plot” for each xj in the model, using the following: Bjxij \+ Ei. You are forcing a situation where if the data is linear, as you want it to be, the plot should look like a straight line. Therefore, if it doesn’t look like a straight line, transformation may be required for one or more of the x’s. For instance, if the data appears to be curved, you may want to add a quadratic term.

 

While we could check for problems in the data using non-visual means, it is generally the case that after identifying a problem, you are going to have to look at some sort of graph to see what’s going on and fix it anyway. Therefore, looking at the graphs described here to check for the desired properties skips past the first step to the inevitable usage of graphs to analyze data.

 

***  


            “No, Sherlock, what you’re saying makes absolutely no sense.”

            Sherlock paused, his curled fingers half-heartedly playing an invisible violin while he seemed to be thinking of a better way of articulating himself. He blinked six times. (John counted.)

            “Well, he entered the room before her, with her mobile in his hand, but if he’d set it down first he could achieve leverage by placing his arm…” Sherlock shook his head, clearing the sentence, “Given her height, the torque resulting from the fact that her feet served an effective fulcrum, and the force of his shove, she would have struck the bed with at least…” He pulled out a table of numbers, and handed it to John. “Just—don’t you see? Their positioning was unstable enough that there would have had to be a struggle, and based on her height she would have fallen in this range of—”

            John gave the paper a six-second glance. (Sherlock counted.) “What I’m _trying_ to say, Sherlock, is can you even _picture_ what’s going on, here? Because I don’t think it’s physically possible in the first place.”

            “I can picture it perfectly,” Sherlock answered, indignantly taking the paper back.

            John chuckled, and then chuckled more at the affronted look Sherlock was giving him. “Then you think your description will stand up to the rigor of scientific testing?”

            “I think any alternative way the Yard attempts to describe the scenario will, in the end, be corrected degree by degree until it matches mine.”

            “Huh,” John smirked. “Didn’t know you equated _Anderson_ with _scientific rigor_.”

            This time, Sherlock quite nearly gasped. He huffed and shot up from his chair, pacing madly. “I do _not_ ,” he said, and then paused and turned on his toes to appraise John with narrowed eyes. “What were you trying to refer to, if not the Yard’s investigations?”

            “Mm,” was all John said—hummed, really—as he closed his laptop and stood up, stepping past Sherlock. He marched into the kitchen and looked over Sherlock’s sundry experiments and equipment, littering the table and making it unusable for the foreseeable future. He spotted one particularly interesting (interesting: a very kind word for _disgusting_ ) beaker of _something_ , which appeared to be less a chemistry experiment gone wrong and more a biology experiment gone far, _far_ too right, plucked it off the table, turned on his heel, and set forth up the stairs.

            “John!” Sherlock snapped. “Put that _back,_ it’s very sensitive to any sort of temperature gra— _John!_ ”

            John had finished making his way up the stairs and stepped into his room, beaker still in hand, apparently completely oblivious to Sherlock’s shouting. Sherlock took the stairs in twos, and then rapped on John’s closed door.  “John, whatever form of revenge this is, it is absolutely _not_ acceptable.”

            He was reasonably certain he heard the faint clinking of John tapping on the glass of the beaker.

            “If you disrupt the growth cycle of the—”

            The clinking grew louder.

            Sherlock threw the door open. “ _John_ —”

            Several things happened in rapid succession. Sherlock would later realize that the moment he had opened the door, John had set the beaker on the nightstand, where it would have been perfectly fine if it were not for the four-dimensional temperature gradient soon to occur (well, currently occurring: the fourth dimension was time). But Sherlock could not have realized this just at the moment of his opening the door, because what happened only a split second afterward was an unreckonable force shoving him from the doorway to the wall, ninety degrees away on the horizontal plane.

            Then, John (John, the unreckonable force) was pressed up against him. Sherlock felt the proximity first in his groin, of all places, pinned against John’s belly, which Sherlock had always imagined would be comfortingly plush but should have known would contain, like John, beneath the fuzzy outer layer, knots and bundles of muscle and fierce, unwavering competence. John’s belly pinned Sherlock against the wall _competently,_ so very effectively.

            Or perhaps that was his hands, one of which was wrapped around Sherlock’s thin wrists and pinning them just above Sherlock’s head, the effort of it forcing John to his tiptoes, forcing John to rub upward against Sherlock; the other of which was gripping Sherlock’s jaw between its thumb and forefinger, forcing his chin up so that he had to strain to look down at John.

            John pulled Sherlock’s face down and leaned heavier against him to whisper in his ear. “I said scientific rigor,” John told him. “I _meant_ a demonstration, not a discussion over whether bloody Scotland Yard was going to figure out the meaning of your convoluted description.”

            Sherlock got the point. Sherlock was reasonably certain he got the point, but was perhaps slightly distracted from it, because somewhere between the sitting room and here John had done it again, that _thing_ , that thing John did, where he walked out of his body and became some sort of otherworldly fire, reeking of gunpowder and spent electricity and danger and walking on water and somehow transcending the world around him while still, by all appearances, belonging in it. John did that _thing_ that nobody else could do, that thing where he continually and abruptly and _forcefully_ surprised Sherlock.

            He thought, probably, that John would back down momentarily, would retreat back into the world, would become jumpers and tea instead of danger and gunpowder, would contain the fire back into a burning ball in his chest, where he kept it well hidden.

            But John maintained his position, angled Sherlock’s head and looked into his eyes. “Do you understand, yet? Or do I need to continue?” he asked, his voice a threatening growl—not low, like Sherlock’s, but deceptively midrange, blending in with the silence around them, distinguishable only by its grinding and crackling, the sound of sparks and popping embers.

            Sherlock understood. From here, John had a formidable amount of control over Sherlock, despite their difference in height. Sherlock had assumed a struggle: he had not assumed pinned wrists, a strong hand on her jaw so that she was seconds away from strangulation. He understood.

            John knew he understood. John asked anyway.

            “Continue,” Sherlock breathed.

            “Right,” John answered, all business, before moving his hand slowly down Sherlock’s throat, resting his fingers gently around it. “As you can see,” he whispered in crackling embers, making sure not to press against Sherlock any more than necessary to demonstrate the positioning of his hand, “no,” John continued; Sherlock considered leaning forward into John’s hand, felt the compulsion to lean forward, and he did, and the tightness of his breathing burned through him like a string of unceasing realizations, prodding reminders of his mortality, his lips parted in an _oh_ ; John pulled his wrists from the wall, “struggle,” he said, and then walked into Sherlock until Sherlock was tripping over his ankles, was forced to fall backward from knees bent by contact with the edge of the bed, “necessary.”

            John held Sherlock’s hands and pinned them back above his head, against the blankets, and climbed onto Sherlock to maintain the light grip of his fingers on Sherlock’s throat. Sherlock bit his lip, but could not force his eyes from John straddling over him, from the faintest, lightest contact of John’s groin against his. He could not sit up for a better view, could not strain too hard against John’s hand on his throat.

            “Of course,” Sherlock finally said through a voice that was more desperate than he expected, “there is one particular difference between your demonstration and what really happened.”

            “Oh?” John tilted his head up, face stony, daring Sherlock to prove him wrong.

            “Yes,” Sherlock muttered. “By this point, the victim had said _no_.”

            “ _Oh_ ,” John sighed, and then groaned, “ _oh,_ ” and slowly, deliberately, tilted his hips forward, and Sherlock could feel, as only a faint, agonizing _itch_ of heat, every inch of John’s hardening cock rubbing into him through their trousers. John released Sherlock’s wrists to cup his hand around his own growing bulge. “ _Fuck_ ,” he hissed as he grabbed harder, and then released his grip to run a hand through his hair.

            Sherlock elevated his own hips, striving to make contact again with John’s erection, coughing as John unconsciously increased pressure on Sherlock’s neck. John slid his hand from Sherlock’s neck and wrapped it around to the back of his head, yanking on his hair, and Sherlock arched up further, finally meeting John, rubbing against John, _John_ , John. “ _John_ ,” he choked, wondered vaguely when his throat had tightened so much, when the corners of his eyes had built up desperate droplets.

            “Jesus _Christ_ ,” John growled, and lowered himself down against Sherlock, elbows on either side of Sherlock’s shoulders so that he could pull Sherlock’s hair back with both hands, expose his throat and trace his lips over it, suction his lips onto it, suck his way over Sherlock’s Adam’s apple and up to his jaw to gnaw on his ear. He ground his hips down hard against Sherlock, breathed out a helpless moan. “You’re _hard_ ,” he muttered while he bit down, while Sherlock squirmed beneath him in the perfect balance of pain and taking the pain out on John by rubbing furiously into him.

            “Surprised?” Sherlock breathed, grabbing John by the shoulders to steady each stroke of his cock against John. John reached down to unbutton and unzip his trousers. “Don’t,” Sherlock rubbed, gasped, rubbed, “bother.”

            “ _Ngh_.” John retreated, curled his fingers back into Sherlock’s hair with vigor, leaned in and closed his teeth around a patch of skin on Sherlock’s throat, groaning and thrusting with progressively jerkier piston-punches of his erection against Sherlock. He pulled back to see Sherlock’s eyes shining and brows knit with desperation, lower lip pinned hard beneath his teeth.

            “ _John,_ ” Sherlock let out a strangled sound, “get me off, John, _please_.”

            “Oh, _god,_ ” John breathed, swallowed. “Yes.” He reached down to rub his hand against his own trousers, shuddered, squared his shoulders, looked back to Sherlock with sharp eyes. “You fuck yourself against this,” he rumbled, grabbing the bulge at the front of his trousers, “and I’ll do the rest.”

            Sherlock whimpered and then set himself to task, pulling John down by the hips to press them together and jerking his own hips frantically against him, harder and harder until there was no pressure, only the maddening closeness of the edge, and John, choking back groan after groan and then finally tilting his head down and closing his mouth over Sherlock’s, moaning into his throat, conveying to him directly the one singular note that was John, the frequency of otherworldly fire, the wavelength of danger, and Sherlock’s hips snapped, snapped, snapped against John and then Sherlock was infused with heat, too, felt it spilling over him and ebbing, felt the fire flare and then retreat as John pressed deep against Sherlock and released one guttural cry.

He watched John gather the heat back into his chest, compact it until it burned a steady glow amidst his heart. John became not ordinary but extraordinary in a different fashion, so much force of will contained within one man with such subtlety that not even Sherlock had seen it, at first.

            “Bloody _hell_ ,” John rolled off of Sherlock. He looked to him with fresh eyes, with clearer eyes, no longer fogged by need. He bit his lip. “Was that…er…”

            “Thank you,” said Sherlock, lacking anything better to say, or anything that John could make sense of, anyway. “I concede to the power of visual demonstrations in verifying my theories.”

            John grinned.

            “I suppose it is unrealistic to expect I might always have a perfect visualization once so many variables are involved.” He tilted his head. “For instance, for the life of me I cannot quite imagine exactly what you might look like without pants on.”

            “I suppose this calls for another demonstration?”

            “Perhaps tomorrow.” Sherlock sat up and moved to the edge of the bed.

            “What are you doing?”

            “Leaving you for your night’s rest,” Sherlock said, standing up and grabbing the glass beaker. “It’s past midnight.”

            “Get back here, you git,” John patted the bed. “You need rest too, and I need to set a good example for you of what a full night’s rest looks like. Maybe you’ve just never seen it before.”

            “But my experiment—” he motioned to the beaker.

            “Is a lost cause,” John stood up and took it from him. “You said so yourself, something about the temperature. Now, come on.”

            “Are you sure you’re quite comfortable with…”

            “Sherlock,” John stared at him pointedly, climbing back onto the bed. “I’ve shared a room with someone for almost my entire life. So help me, if you think that sharing a bed with you is the worst thing to ever happen to me, you have not spent eight years listening to Harry snore in the bunk above you.”

            “I…”

            “I sleep better with someone nearby; I _am_ better with you nearby; therefore, by sitting your arse down and getting under the covers you should be granting me the best sleep of my life. You’re doing _me_ a favor.” John peeled the blankets back.       Sherlock sat back down on the edge of the bed, curled his legs up and dipped his toes under the covers tentatively.  “I’d also like to, er…I mean, since we just…I know it’s _weird_ , but…” John inhaled, exhaled resolutely. “How do you feel about cuddling?”

            Sherlock pulled the blankets up around him, and the corners of his mouth quirked into a smile. “Allow me to demonstrate.”


End file.
